Ch 8: ExponentsA) Product & Power Properties

Objective:

To recognize the properties of exponents and use them to simplify expressions.

x3

x x x=

exponent

base

BaseThe foundation of an expression that is raised to a power is known as the base.ExponentAn exponent represents the number of times an expression is multiplied by itself and is written in superscript.

Definitions

n n n n n

xa

=xb xa + b

n3

=n2 n3 + 2

Example:

= n5n5

1) Expand the bases as many times as the exponents states.2) Count the number of times the variable appears – that is

the exponent.Shortcut: Evaluate the bases separately and ADD the exponents.

Product Property Rules

Example 1 Example 2

Example 3 Example 4

x2 x3

= x5

2x2 3x3

= 6x5

(2x)2 3x3

= 12x5

2xy2 −3x3y

= −6x4y312 x5 −6 x4 y3

Classwork

1)

2)

3)

4)

5)

6)

€

x4 • x3

€

k5 • k6

€

3m2 • m5

€

2x3 • 4x5

€

−3n7(2n2 )

€

−3a4 (−2a2 )

2 xxx 4 xxxxx = 8x8

-3 nnnnnnn 2nn = -6n9

-3 aaaa -2aa = 6a6

Example:

1) Expand the base (whatever is inside the parenthesis) as many times as the exponent states.

2) Continue expanding until there are no more exponents.3) Count the number of times the variable appears – that is

the exponent.Shortcut: MULTIPLY the exponents.

( )xa

=b

xab

( )n3

(n ) (n )3

= n32

nnn nnn = n6

2

3

Power to Power Property Rules

Example 5 Example 6

Example 7 Example 8

(x2)3

= x6

(2x2)3

= 8x6

(-2x2)3

= -8x6

(2x2y)3

= 8x6y3

Classwork

1)

3)

2)

4)

€

(y4 )5

€

(−4x2y)2

€

(2a2b3c5 )3

€

−23

a4b2 ⎛ ⎝

⎞ ⎠

2

€

y20

€

16x4y2

€

8a6b9c15

€

49

a8b4

€

(3x2y3 )2 (−4xy2)2

€

144x6y10

€

9x4y6 • 16x2y4

5) 6)

€

(2a3b5c)2 (−2ab4 )3

€

−32a9b22c2

€

4a6b10c2(−8a3b12)